**TEMPEST** (Tennessee Electromagnetics/Fluids/Plasmas Equation Solver Toolchain) is our high-performance software test-bed for developing plasma simulations. It was written by Dr. Joel Thompson at UTSI during his doctoral work. It has since been expanded and improved. The software is capable of simulating unsteady or steady flows in multidimensional, complex geometries with a number of physical models and additional multiphysics coupling modules. It has been parallelized to make maximum use of high-performance computing clusters. A source-controlled repository is maintained for ongoing simulation work.

## Overview

A quick overview of the major points of the **TEMPEST** software is:

- Capable of 1D, 2D or 3D simulations
- Solves viscous, compressible Navier-Stokes equations coupled to either electrostatic Poisson equation, magnetic diffusion equation (resistive MHD) or full Maxwell equations
- Fully unstructured meshing; mixed elements with any number of sides is permitted
- Very robust geometry module permits easy handling of complicated domains in a very generalized manner
- High-order accurate time integration schemes including Runge-Kutta explicit or pseudotime implicit integration
- High-order accurate gradient reconstruction using either node- or face-based Green-Gauss or least-squares methods
- Node and face data interpolation permits the construction of higher-order schemes
- Both Riemann-solver-based and flux-splitting-based shock-capturing schemes

## Electromagnetic Coupling

Current capabilities for the coupling between the fluid and electromagnetic systems include:

- Coupling between the Navier-Stokes and full Maxwell equations, including conduction, convective and displacement current effects, low-electrical-conductivity behavior, non-neutral electric charge separation effects, and electrode edge effects
- Resistive magnetohydrodynamic (magnetic diffusion approximation) model
- Coupling between the Navier-Stokes and electrostatic Poisson equation (electrohydrodynamic model)

## Time Integration Schemes

**TEMPEST** boasts several explicit and implicit time integration schemes, including:

- Explicit RK2, RK3, RK4 and user-defined adjustable midpoint RKn methods, which provide between second- and fourth-order time accurate solutions
- Pseudotime point-implicit relaxation with arbitrary second- to sixth-order accuracy
- Pseudotime implicit relaxation with RK4-explicit subadvance
- Common predictor-corrector methods, including Hancock and MacCormack

## Numerical Flux Calculation

Several numerical flux modules have been added, including:

- Roe approximate Riemann solver using strongly conservative eigendecomposition for Navier-Stokes/Maxwell equations
- Source-coupled HLL, HLLC, Rusanov, AUSM+ fluid approaches and Roe Maxwell solver

## Higher-order Accuracy

Higher-order accuracy can be provided by gradient reconstruction modules that have been added, including:

- Node-based or face-based Green-Gauss gradient reconstruction
- Least-squares gradient reconstruction
- Interpolation of cell-centered data to nodes and faces between timesteps
- Pseudotime preconditioning to accelerate convergence

## Gallery of Tests

### Radio Wave Propagation and Diffusion

A step magnetic field is forced on the left of a one-dimensional region of meter length. Depending on the electrical conductivity of the region, the discontinuity will either propagate as a simple wave (for zero conductivity), or be diffused (large conductivity begins to show the MHD limit being reached). In the three graphs below, different conductivities were tested. Leftmost is zero conductivity (wave propagation of the electromagnetic field), middle graph has a very small conductivity of 0.01mho/m, and last graph has a higher conductivity of 0.1mho/m. Green lines in the last two graphs indicate the analytical solution according to MHD theory. In the first and second graph, the MHD theory fails to capture the correct physical behavior. **TEMPEST**, however, correctly renders both wave and MHD behavior. In the last simulation with 0.1mho/m, **TEMPEST** agrees with MHD theory to within 1.5%.

### Orszag-Tang MHD Turbulence

The Orszag-Tang MHD problem is a common 2D MHD turbulence problem used to test MHD solvers. In this problem, **TEMPEST** was applied without the MHD solver, but instead solving the coupled Navier-Stokes and full Maxwell equations. This means that no eigenvalues or eigenvectors from MHD theory were applied; only the fully coupled form of the fluid and Maxwell equations. The result below shows the difference between the AUSM (left) and Roe (right) methods, which agree very well to each other and to the classic MHD solution.

### MHD Kelvin-Helmholtz Instability

A Kelvin-Helmholtz instability was set up on a periodic square grid, with a right-moving high-density column of fluid, and a left-moving low-density column of fluid. Three cases were run: the top graph corresponds to the purely hydrodynamic case, where the electrical conductivity was zero. Here, the electromagnetic fields propagated and did not interact with the fluid. In the second case, a very small electrical conductivity was run. Slight suppression of the low-wavelength instabilities is seen. In the final case, a very high electrical conductivity was applied, resulting in an MHD case, where the low-wavelength instabilities were completely suppressed.